A modified form of Siegel's mean-value theorem

Abstract

The Minkowski–Hlawka theorem† asserts that, if S is any n -dimensional star body, with the origin o as centre, and with volume less than 2ζ( n ), then there is a lattice of determinant 1 which has no point other than o in S . One of the methods used to prove this theorem splits up into three stages, ( a ) A function ρ (x) is considered, and it is shown that some suitably defined mean value of the sum taken over a suitable set of lattices Λ of determinant 1, is equal, or approximately equal, to the integral over the whole space. ( b ) By taking ρ (x) to be equal, or approximately equal, to where σ(x) is the characteristic function of S , and μ( r ) is the Möbius function, it is shown that a corresponding mean value of the sum where Λ* is the set of primitive points of the lattice Λ, is equal, or approximately equal, to